Functional equations of prehomogeneous zeta functions and intertwining operators

Functional equations of prehomogeneous zeta functions and intertwining operators

Fumihiro Sato

§0 Introduction

In the present paper, we establish a relation between the gamma matrices of the func- tional equations satisfied by zeta functions associated with prehomogeneous vector spaces and certain integrals related to the intertwining operator of degenerate principal series representations of GLm.

In [S2] and [S3] we proved that the functional equation of the zeta functions associated with a prehomogeneous vector space can be formulated as the functional equation satisfied by O(m)-invariant spherical functions on a certain weakly spherical homogeneous space acted on by GLm. The spherical functions are defined to be the Poisson transforms of distributions on the weakly spherical homogeneous space which belong to a degenerate principal series representation of GLm and the explicit form of the functional equation satisfied by them can be determined if we can calculate the images of the distributions under the intertwining operator. This gives a new method of calculating the gamma matrices of the functional equations of prehomogeneous zeta functions.

In §1 we formulate our main result (Theorem 1), which expresses the gamma matrix in terms of an integral of a complex power of certain polynomial obtained as a special- ization of the fundamental relative invariant of the prehomogeneous vector space under consideration. Some concrete examples are discussed in §2. The proof of the main result is given in §3.

As the examples in §2 show, Theorem 1 often simplifies the calculation of the gamma matirces. However the explicit evaluation of the integral I_ij(λ) in Theorem 1 seems quite difficult for prehomogeneous vector spaces with complicated fundamental relative invariant. Therefore our interest in the main theorem is of theoretical nature. We shall discuss some theoretical implication of our approach in the final remark in §3.

The result of the present paper can be generalized to the functional equations satisfied by the p-adic local zeta functions associated with prehomogeneous vector spaces.

§1 Main result

Let m and n be positive integers with m > n. Let H be an R-subgroup of GL_m with no R-rational characters and V =M_m,n the space of m by n matrices. Then the group H×GL_n acts linearly on the matrix space V by

ρ(h1, h2)(v) = h1v^th2 (h1 ∈H, h2 ∈GL_n, v ∈V).

We assume that (H×GL_n, ρ,V) is a regular prehomogeneous vector space whose singular set Sis anR-irreducible hypersurface. Denote by f(v) the fundamental relative invariant of (H×GL_n, ρ,V) over R and by f^∗(v^∗) the fundamental relative invariant of the dual prehomogeneous vector space (H×GL_n, ρ^∗,V^∗) overR. In the following, we identify V^∗ with V = M_m,n by the inner product hv, v^∗i = tr(^tvv^∗). Then we have ρ^∗(h₁, h₂)(v^∗) =

th⁻¹₁ v^∗h⁻¹₂ . Put d = degf(v) and d^∗ = degf^∗(v^∗). We denote by S^∗ the singular set of (H×GL_n, ρ^∗,V^∗).

In the following, we put

H =H(R), GLn =GL_m(R), V =V^∗ =M_m,n(R), S =S(R), S^∗ =S^∗(R).

LetV −S =V1∪ · · · ∪Vν and V^∗−S^∗ =V₁^∗∪ · · · ∪V_ν^∗ be the decomposition ofV −S and V^∗−S^∗ into connected components. For i, j (1≤i, j ≤ν) ands∈C with Res ≥0, we put

|f(v)|^si =

(|f(v)|^s (v ∈Vi),

0 (v 6∈Vi), |f^∗(v^∗)|^sj =

(|f^∗(v^∗)|^s (v^∗ ∈V_j^∗), 0 (v^∗ 6∈V_j^∗).

We denote also by the same symbols|f(v)|^si and |f^∗(v^∗)|^sj the distributions on V and V^∗, respectively, defined by the analytic continuations of the functions above. We define the local zeta functions Φi(φ;λ), Φ^∗_i(φ^∗;λ) (1≤ i ≤ ν, φ ∈ S(V) and φ^∗ ∈ S(V^∗)) attached to these prehomogeneous vector spaces by

Φi(φ;λ) = Z

V |f(v)|^{(λ−mn/2)/d}i φ(v)dv, Φ^∗_j(φ^∗;λ) = Z

V^∗|f^∗(v^∗)|^{(λ−mn/2)/d}j ^∗φ^∗(v^∗)dv^∗. Denote by ˆφ^∗ the Fourier transform of φ^∗ ∈ S(V^∗):

φˆ^∗(v) = Z

V^∗

φ^∗(v^∗) exp(2π√

−1hv, v^∗i)dv^∗.

Then, by the general theory ([SS], [S1]), the following functional equation holds:

(1.1) Φi( ˆφ^∗;λ) =

ν

X

j=1

γij(λ)Φ^∗_j(φ^∗;−λ),

where γij(λ) are meromorphic functions of λ independent of φ^∗ and have elementary expressions in terms of the Γ function and exponential functions.

The main result of this paper is the following thoerem which gives an expression of γij(λ) as an Eulerian integral (an integral of a complex power of a polynomial).

Theorem 1 Put v₀^∗ = 0

En

∈V^∗ and, for each j (1≤j ≤ν), take agj ∈SL_m(R) such that v_j^∗ =^tg_j⁻¹v^∗₀ ∈V_j^∗ and

f^∗(v^∗_j)

= 1. Then we have γij(λ) = π−λ−(m−n)n/2Iij(λ)

n−1

Y

k=0

Γ _2n^λ +^m−2k₄ Γ

−2n^λ −^m−2(n−k)4

, where

Iij(λ) = Z

Mm−n,n(R)

f

gj

x En

(λ−mn/2)/d i

dx.

Remarks. (1) The integral Iij(λ) may be divergent for any λ (cf. §2 Example 1).

Then we understand Iij(λ) to be the value at α = 0 of the function Iij(α, λ) defined by the analytic continuation of the integral

Iij(α, λ) = Z

Mm−n,n(R)

f

gj

x E_n

(λ−mn/2)/d i

det En+^txx−α

dx, which is absolutely convergent when Re(λ)> ^mn₂ and Re(α) is sufficiently large.

(2) As mentioned in the introduction, the integral Iij(λ) is related to an integral defining an intertwining operator of degenerate principal series representation of GL_m. The relation will be clarified in§3.

§2 Calculation of gamma matrices

In this section, we give some samples of the calculation of the gamma matrices γij(λ) based on Theorem 1.

Example 1: The case of indefinite quadratic forms.

Let m be a positive integer greater than 1 and p, q positive integers with p+q = m.

Denote by SO(p, q) the special orthogonal group of the symmetric matrix Ep,q =

Ep 0 0 −Eq

.

The group SO(p, q)×GL₁ acts on the vector space M_m,1 by

ρ(h, t)·v =thv (h∈SO(p, q) t∈GL₁, v ∈M_m,1)

and the triple (SO(p, q)×GL₁, ρ,M_m,1) is a regular prehomogeneous vector space with fundamental relative invariant f(v) = ^tvEp,qv. The dual prehomogeneous vector space (SO(p, q)×GL₁, ρ^∗,M_m,1) is given by

ρ^∗(h, t)v^∗ =t^−1th⁻¹v^∗ (h∈SO(p, q)t ∈GL₁, v^∗ ∈M_m,1) Its fundamental relative invariant is f^∗(v^∗) =^tv^∗Ep,qv^∗. For i, j = 1,2, we put

Vi =

v ∈M_m,1(R) =R^m

sgnf(v) = (−1)ⁱ⁻¹ , V_j^∗ =

v^∗ ∈M_m,1(R) = R^m

sgnf^∗(v^∗) = (−1)^j−1 .

Since the degree of the fundamental relative invariants is equal to 2, we have, for φ, φ^∗ ∈ S(R^m),

Φi(φ;λ) = Z

V

^tvEp,qv

(2λ−m)/4

i φ(v)dv,

Φ^∗_j(φ^∗;λ) = Z

V^∗

^tv^∗Ep,qv^∗

(2λ−m)/4

j φ^∗(v^∗)dv^∗.

The explicit form of the functional equation (1.1) is well-known in this case (see, e.g., Gelfand-Shilov [GS]):

Proposition 2 For i= 1,2, we have Φi( ˆφ^∗;λ) = X

j=1,2

γij(λ)Φ^∗_j(φ^∗;−λ),

where

γ11(λ) γ12(λ) γ₂₁(λ) γ₂₁(λ)

= π^−(λ+1)Γ

2λ−m+ 4 4

Γ

2λ+m 4

× sinπ ^p−q−2λ₄

sin^pπ₂ sin^qπ₂ sinπ ^q−p−2λ₄

! . Put v₀^∗ =^t(0, . . . ,0,1)∈R^m. Then, we can take

g₁ =





0

¹

·· ·

1

0



, g₂ = 1, v₁^∗ =^t(1,0, . . . ,0)∈V₁^∗, v^∗₂ =v^∗₀ ∈V₂^∗.

Hence Iij(λ) =









 Z

R^m−1

1 + (x²₁+· · ·+x²_p−1)−(x²_p+· · ·+x²_m−1)

(2λ−m)/4

i dx (j = 1), Z

R^m−1

(x²₁+· · ·+x²_p)−(x²_p+1+· · ·+x²_m−1)−1

(2λ−m)/4

i dx (j = 2).

Then we can obtain the following proposition, from which Proposition 2 follows immedi- ately.

Proposition 3

I11(λ) I12(λ) I₂₁(λ) I₂₂(λ)

= π^(m−3)/2·Γ

2−m−2λ 4

Γ

2λ−m+ 4 4

× sinπ ^p−q−2λ₄

sin^pπ₂ sin^qπ₂ sinπ ^q−p−2λ₄

! .

Proof. Since the calculation is quite similar, we prove the formula only forIi1(λ). We put

|x|^s+=

(|x|^s (x >0),

0 (x≤0), |x|^s− =|−x|^s+. Then we have

I_i1(λ) = π^(m−1)/2 Γ(^p−1₂ )Γ(^q₂)

Z ∞ 0

Z ∞

0 |1 +r−t|^(2λ−m)/4± r^{(p−1)/2−1}t^q/2−1dr dt

= π^(m−1)/2 Γ(^p−1₂ )Γ(^q₂)

Z ∞

0 |1 +r|^{(2λ−p+q)/4}r^{(p−1)/2−1}dr Z ∞

0 |1−t|^(2λ−m)/4± t^q/2−1dt.

Here we take the + sign for i = 1 and the − sign for i = 2. By the well-known formula for the Euler B-function, we easily have

Z ∞

0 |1 +r|^{(2λ−p+q)/4}r^{(p−1)/2−1}dr = Γ(^−2λ−m+2₄ )Γ(^p−1₂ ) Γ(^−2λ+p−q₄ ) , (2.1)

Z ∞

0 |1−t|^(2λ−m)/4± t^q/2−1dt =









 Z

0 1

(1−t)^(2λ−m)/4t^q/2−1dt (i= 1) Z

1

∞

(t−1)^(2λ−m)/4t^q/2−1dt (i= 2) (2.2)

=















Γ(^2λ−m+4₄ )Γ(^q₂)

Γ(^2λ−p+q+4₄ ) (i= 1)

Γ(^−2λ+p−q₄ )Γ(^2λ−m+4₄ )

Γ(1− ^q₂) (i= 2).

Combining these identities, we obtain the proposition.

Remark. Note that there exist no λ’s for which both the integrals (2.1) and (2.2) converge absolutely. However the calculation can be justified by the method indicated in Remark just after Theorem 1.

Example 2: The case of the discriminant of binary cubic forms.

Let us consider the prehomogeneous vector space (SL2×GL₁, ρ,M_4,1), in which the group SL₂ acts onM_4,1via the 3rd symmetric tensor representationS³. LetH=S³(SL₂) be the image of SL₂ in GL₄. We identify the vector space M_4,1 with the space of homogeneous polynomials in the variables U, V of degree 3 by

v =







 x1

x2

x3

x₄







7−→x1U³+x2U²V +x3UV² +x4V³.

Then the fundamental relative invariant f(v) is given by the discriminant of the corre- sponding cubic form:

f(v) = x²₂x²₃+ 18x1x2x3x4−4x1x³₃−4x³₂x4−27x²₁x²₄. Since we have

tH⁻¹ =







 1

3 3

1









−1

H







 1

3 3

1









in GL4, the fundamental relative invariant f^∗ of the dual prehomogeneous vector space is given by

f^∗(^t(y1, y2, y3, y4)) =f(^t(y1,3y2,3y3, y4)) = 3³ 3y₂²y²₃+ 6y1y2y3y4−4y1y₃³−4y₂³y4−y₁²y²₄ . For i, j = 1,2, we put

Vi =

v ∈M_4,1(R) =R⁴

sgnf(v) = (−1)ⁱ⁻¹ , V_j^∗ =

v^∗ ∈M_4,1(R) = R⁴

sgnf^∗(v^∗) = (−1)^j−1 .

Since the degree of the fundamental relative invariants is equal to 4, we have, for φ, φ^∗ ∈ S(R⁴),

Φi(φ;λ) = Z

V |f(v)|^(λ−2)/4i φ(v)dv, Φ^∗_j(φ^∗;λ) =

Z

V^∗|f^∗(v^∗)|^(λ−2)/4j φ^∗(v^∗)dv^∗.

In this case the explicit form of the functional equation (1.1) was calculated by Shintani ([Sh]):

Proposition 4 For i= 1,2, we have Φi( ˆφ^∗;λ) = X

j=1,2

γij(λ)Φ^∗_j(φ^∗;−λ),

where

γ11(λ) γ12(λ) γ21(λ) γ21(λ)

= 3^(3λ+2)/2

2 ·π^−(λ+2)Γ λ

4 +1 2

2

Γ λ

4 + 1 3

Γ

λ 4 + 2

3

× −sin ^πλ₂

cos ^πλ₄ 3 cos ^πλ₄

−sin ^πλ₂

! .

Shintani’s proof is quite involved and far from trivial. As we shall see below, Theorem 1 gives a more straightforward way to the proof of Proposition 4.

Proof of Proposition 4. Put

v^∗₀ =







 0 0 0 1









and gj =









1 0 0 0

0 1 0 0

0 0 1 0

0 (−1)^j−1 0 1

















(2²3³)^−1/4

0

1

0

1 ₍₂²₃³₎^1/4







 .

Then, v_j^∗ = ^tg_j⁻¹v₀^∗ = (2²3³)^−1/4·^t(0,(−1)^j,0,1)∈ V_j^∗ and

f^∗(v_j^∗)

= 1. Hence, putting ǫ= (−1)ⁱ⁻¹, η = (−1)^j−1, we have

I_ij(λ) = (2²3³)^(λ+2)/4 Z

R³

f ^t(x₁, x₂, x₃,1 +ηx₂)

(λ−2)/4

ǫ dx₁dx₂dx₃. Since

f(v) = −27x²₄ (

x1−9x2x3x4 −2x³₃ 27x²₄

2

− 2²(x²₃−3x2x4)³ 3⁶x⁴₄

) ,

we obtain

Iij(λ) = 2^(λ+2)/23^3λ/2 Z

R³|1 +ηx2|^(λ−2)/2

x²₁− 2²(x²₃−3x2(1 +ηx2))³ 3⁶(1 +ηx2)⁴

(λ−2)/4

−ǫ

dx1dx2dx3

= 2^(λ+2)/23^3λ/2X

τ=±

Z

R³

|1 +ηx₂|^(λ−2)/2 2²|x²₃−3x2(1 +ηx2)|³τ

3⁶(1 +ηx2)⁴

!(λ−2)/4+1/2

×

x²₁−τ

(λ−2)/4

−ǫ dx1dx2dx3

= 2^λ+1/2 X

τ,µ,ν=±

Z

R²

|1 +ηx2|^−(λ+2)/2µ (3|x2|ν|1 +ηx2|µ)^3λ/4+1/2

x²₃ −µν

3λ/4

τ dx2dx3

× Z

R

x²₁ −τ

(λ−2)/4

−ǫ dx1

= 2^λ+13^(3λ+2)/4 X

τ,µ,ν=±

Z

R|1 +x2|^(λ−2)/4µν |x2|^3λ/4+1/2ην dx2

× Z

R

x²₃ −µ

3λ/4 τ dx3

Z

R

x²₁−τ

(λ−2)/4

−ǫ dx1.

By the formula for the Euler B-function, the integrals can be easily calculated and we have

X

ν=±

Z

R

|1 +x2|^(λ−2)/4µν |x2|^3λ/4+1/2ην dx2 = Γ

λ+ 2 4

Γ

3λ+ 6 4

Γ (λ+ 2) ·c⁽¹⁾_µ,η(λ),

Z

R

x²₃−µ

3λ/4

τ dx3 = √

π· Γ

3λ+ 4 4

Γ

3λ+ 6 4

·c⁽²⁾_τ,µ(λ),

Z

R

x²₁−τ

(λ−2)/4

−ǫ dx1 = √

π· Γ

λ+ 2 4

Γ

λ+ 4 4

·c⁽³⁾_ǫ,τ(λ),

where

C⁽¹⁾(λ) = c⁽¹⁾_µ,η(λ)

µ,η=± =













sin ^πλ₄

cos ^πλ₂ 1 1 sin ^πλ₄

cos ^πλ₂













C⁽²⁾(λ) = c⁽²⁾_τ,µ(λ)

τ,µ=± =







− 1

cos ^3πλ₄

sin ^3πλ₄ cos ^3πλ₄

1 0







C⁽³⁾(λ) = c⁽³⁾_ǫ,τ(λ)

ǫ,τ=± =







1 0

− 1

sin ^πλ₄ −cos ^πλ₄ sin ^πλ₄





. The calculation above shows that

I11(λ) I12(λ) I21(λ) I22(λ)

= 2^λ+1/23^(3λ+2)/4 ·π· Γ

λ+ 2 4

2

Γ

3λ+ 4 4

Γ (λ+ 2) Γ

λ+ 4 4

·C⁽³⁾(λ)C⁽²⁾(λ)C⁽¹⁾(λ).

Hence we have I11(λ) I12(λ)

I₂₁(λ) I₂₂(λ)

= 2^λ+1/23^(3λ+2)/4 ·π· Γ

λ+ 2 4

2

Γ

3λ+ 4 4

Γ (λ+ 2) Γ

λ+ 4 4

·













sin ^πλ₂

cos ^πλ₂ −cos ^πλ₄ cos ^πλ₂

−3· cos ^πλ₄ cos ^πλ₂

sin ^πλ₂ cos ^πλ₂











 .

By Theorem 1, this implies Proposition 4. We note only that the proof requires the identities

Γ(2z) = (2π)^−1/22^−12+2zΓ(z)Γ

z+1 2

, Γ(3z) = (2π)⁻¹3^−12+3zΓ(z)Γ

z+ 1

3

Γ

z+2 3

.

§3 Proof of Theorem 1

In [S2] and [S3], we proved that the functional equation (1.1) can be interpreted as a relation between O(m)-invariant spherical functions on the homogeneous space X = H\GLm. The point is to consider the family

(g⁻¹Hg×GL_n, ρ,M_m,n)

Hg ∈H\GLm

of prehomogeneous vector spaces and to regard the associated zeta functions as functions on the parameter spaceH\GL_m. Namely we consider the functions

H\GLm ∋Hg 7−→ Φi(Hg, φ;λ) :=

Z

V |f(gv)|^{(λ−mn/2)/d}i φ(v)dv, H\GL_m ∋Hg 7−→ Φ^∗_j(Hg, φ^∗;λ) :=

Z

V^∗

f^∗(^tg⁻¹v^∗)

(λ−mn/2)/d^∗

j φ^∗(v^∗)dv^∗. Put

v0 = En

0

, v₀^∗ = 0

En

.

Lemma 5 Assume that φ and φ^∗ are left O(m)-invariant. Then we have Φi(Hg, φ;λ) = Zmn(φ;λ)ωi(Hg;λ),

Φ^∗_j(Hg, φ^∗;λ) = Zmn(φ^∗;λ)ω_j^∗(Hg;λ).

Here we put

Zmn(φ;λ) = Z

{v∈Mm,n(R)|rankv=n}

det(^tvv)(λ−mn/2)/2nφ(v)dv, ωi(Hg;λ) =

Z

O(m)|f(gkv0)|^{(λ−mn/2)/d}i dk, ω_j^∗(Hg;λ) =

Z

O(m)

f^∗(^tg⁻¹kv₀^∗)

(λ−mn/2)/d^∗

j dk,

where dk is the normalized Haar measure on the orthogonal group O(m).

It is not difficult (and well-known) to calculateZmn(φ;λ) forφ0(v) := exp 2π√

−1tr(^tvv) : Zmn(φ0;λ) =π^{−(2λ−mn)/4}

n−1

Y

k=0

Γ _2n^λ + ^m−2k₄ Γ ^m−k₂ .

By this identity as well as Lemma 5, the functional equation (1.1) can be rewritten as (3.1) ωi(Hg;λ) =|detg|⁻ⁿΓmn(λ)

ν

X

j=1

γij(λ)ω_j^∗(Hg;−λ),

where

(3.2) Γmn(λ) =π^λ

n−1

Y

k=0

Γ −2n^λ + ^m−2k₄ Γ _2n^λ + ^m−2k₄ .

Thus the determination of γij(λ) is equivalent to the determination of an explicit formula for the functional equation (3.1).

Let P = Pn,m−n (resp. P^∗ = Pm−n,n) be the upper triangular maximal parabolic subgroup of GLm corresponding to the partition m=n+ (m−n) (resp. m= (m−n) + n). Denote by B(GL_m/P;z₁, z₂) the space of hyperfunctions ψ on GL_m satisfying the condition

(3.3) ψ(gp) =|detp₁|^{z1−(m−n)/2}|detp₂|^z2+n/2ψ(g)

∀p=

p1 ∗ 0 p2

∈P =P_n,m−n . The left regular representation of GL_m onB(GL_m/P;z₁, z₂) defines a representation be- longing to degenerate principal series. We denote also byB(GLm/P;z1, z2)^H the subspace of H-invariant functions.

For z = (z₁, z₂)∈C², we put

Ψz,i(g) =|detg|^z2+n/2|f(gv0)|^n(zi ^{1−z2−m/2)/d}. Then Ψ_z,i(g) belongs to B(GL_m/P;z₁, z₂)^H. Similarly, if we put

Ψ^∗_z,j(g) =|detg|^z1−n/2

f^∗(^tg⁻¹v^∗₀)

n(z1−z2−m/2)/d^∗

j ,

then Ψ^∗_z,j(g) belongs to B(GL_m/P^∗;z₁, z₂)^H.

Denote by A(GLm/O(m)) the space of real analytic functions on GLm/O(m). For any ψ ∈ B(GLm/P;z1, z2), we define its Poisson transform by

Pψ(g) = Z

O(m)

ψ(gk)dk.

Then it is known thatPψbelongs toA(GLm/O(m)) and the image of theGLm-equivariant mapping

P :B(GLm/P;z1, z2)→ A(GLm/O(m))

is characterized by certain differential equations (see [K²MO²T] and [O]). Moreover the mapping P is injective for generic z ∈C².

The images of Ψz,i and Ψ^∗_z,j under the Poisson transform are related to the spherical functions ωi and ω^∗_j as follows:

(3.4)

( PΨz,i(g) =|detg|^z2+n/2ωi(Hg;n(z1−z2)), PΨ^∗_z,j(g) =|detg|^z1−n/2ω_j^∗(Hg;n(z1−z2)).

Now we consider the intertwining operator

T_z :B(GL_m/P;z₁, z₂)−→ B(GL_m/P^∗;z₂, z₁).

It is known that the operatorTzdepends onzmeromorphically. For a continuous function ψ ∈ B(GLm/P;z1, z2),Tzψ is given by

(3.5) Tzψ(g) = Z

Mm−n,n(R)

ψ g

X Em−n

En 0

det En+^txx−α

dx α=0

.

The right hand side is understood to be the value at α = 0 of the analytic continuation of the function defined by the integral. We have the following commutative diagram:

(3.6)

B(GLm/P;z1, z2) −−−→ A^P (GLm/O(m))

Tz



 y





y^×c(z1−z2) B(GLm/P^∗;z2, z1) −−−→ A^P (GLm/O(m)), where the right vertical arrow is the multiplication by

c(z₁ −z₂) = Z

M_m−n,n(R)

det E_n+^txx(z¹−z²−m/2)/2

dx (3.7)

= π^(m−n)n/2

n−1

Y

k=0

Γ

z²−z¹

2 −^m−2(n−k)4

Γ ^z2−z₂ ¹ + ^m−2k₄ .

Now we rewrite the functional equation (3.1) by using (3.4). Then we have PΨ_(z1,z²),i(g) = Γ_mn(λ)

ν

X

j=1

γ_ij(λ)PΨ^∗_(z2,z1),j(g),

where we put λ=n(z1−z2). By the commutativity of the diagram (3.6), this implies the identity

PTzΨ_(z1,z²),i(g) =c(z1−z2)Γmn(λ)

ν

X

j=1

γij(λ)PΨ^∗_(z2,z1),j(g).

Since P is injective on B(GLm/P^∗;z2, z1) for generic (z1, z2) and the both side of the identity above are meromorphic, we obtain

(3.8) TzΨ(z1,z2),i(g) =c(z1−z2)Γmn(λ)

ν

X

j=1

γij(λ)Ψ^∗_(z2,z¹),j(g).

Therefore, if we choose gj as in Theorem 1, then

I_ij(λ) =T_zΨ_(z1,z²),i(g_j) = c(z₁−z₂)Γ_mn(λ)γ_ij(λ).

By (3.2) and (3.7), this proves Theorem 1.

Remark. (1) If one can prove (3.8) directly, namely, without assuming (1.1), the argument can be reversed to give an altenative proof of the functional equations of preho- mogeneous zeta functions. A sufficient condition is that the subspace B(GLm/P^∗;z2, z1)^H is spanned by Ψ^∗_(z2,z1),j (j = 1, . . . , ν). We note that this condition is closely related to the H-orbit structure of GL_m/P^∗.

(2) In the above argument the role in the proof of functional equations played by the Fourier transform is now played by the intertwining operator. To appreciate the meaning of this fact, let us consider the prehomogeneous vector space (SL_r×SL_r×GL₁, ρ,M_r).

Here the representation ρ is given by ρ(h1, h2, t)v = th1v^th2. Then m = r², n = 1, and H =SL_r×SL_r embedded in GL_r². Then, the intertwining operator on the degenerate principal series with respect to the parabolic subgroupP_1,r²₋₁ plays the role of the Fourier analysis on the matrix space M_r in the earlier approach to functional equations. This relation between the usual formulation in the theory of prehomogeneous vector spaces and the present one corresponds precisely the relation between the Godement-Jacquet construction of the standard L-functions of GLn ([GJ]) and the Piatetski-Shapiro-Rallis construction ([PS]). Thus the result above (and the result in [S2] and [S3]) may be viewed as an analogue of the Piatetski-Shapiro-Rallis construction for prehomogeneous zeta functions.

References

[GJ] R. Godement and H. Jacquet: “Zeta functions of simple algebras”, Lecture Notes in Math. 260, Springer-Verlag, 1972.

[GS] I. M. Gelfand and G. I. Shilov: Generalized functions, Vol. I, Academic Press, New York, 1964.

[K²MO²T] M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima and M. Tanaka: Eigenfunctions of invariant differential operators on a symmetric space, Ann. Math. 107(1978), 1–39.

[O] T. Oshima: Generalized Capelli identities and boundary value problems for GL(n), in “Structures of solutions of differential equations, Katata/Kyoto 1995”, Ed. M. Morimoto and T. Kawai, World Scientific, 307–335.

[PS] I. Piatetski-Shapiro and S. Rallis: L functions for the classical groups, notes prepared by J. Cogdell, in “Explicit constructions of automorphic L-functions”, Lecture Notes in Math. 1254, Springer-Verlag, 1987.

[S1] F. Sato: Zeta functions in several variables associated with prehomogeneous vector spaces I: Functional equations, Tˆohoku Math. J. 34(1982), 437–483.

[S2] F. Sato: Eisenstein series on weakly spherical homogeneous spaces and zeta functions of prehomogeneous vector spaces, Comment. Math. Univ. St. Pauli, 44(1995), 129–150.

[S3] F. Sato: Eisenstein series on weakly spherical homogeneous spaces of GL(n), Tˆohoku Math. J. 50(1998), 23–69.

[SK] M. Sato and T. Kimura: A classification of irreducible prehomogeneous vector spaces and their invariants, Nagoya Math. J.65(1977), 1–155.

[SS] M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces. Ann. of Math. 100(1974), 131–170.

[Sh] T.Shintani, On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24(1972), 132–188.

Fumihiro Sato

Department of Mathematics Rikkyo University

sato@rkmath.rikkyo.ac.jp